Let I_ n = ₀¹ e^ -y y^ n d y , where n is a non-negative integer. Then _ n =1 ^ I _ n n ! is

Step-by-step solution

I_ n = ₀¹ e^ -y y^ n d yI_ n =- 1 e +n I_ n-1 (by reduction formula) I_ n -n I_ n-1 =- 1 e array l I_ n n ! - I_ n-1 n-1 ! =- I e(n !) n=1 I₁ 1 ! - I₀ 0 ! =- 1 e n=2 I₂ 2 ! - I₁ 1 ! =- 1 e(2 !) n=3 I₃ 3 ! - I₂ 2 ! =- 1 e(3 !) n=4 I₄ 4 ! - I₃ 3 ! =- 1 e(4 !) eq. (1)+e q (2) 2+ eq (3) 3+( eq. 4) 4+ - I₁ 1 ! + I₂ 2 ! + I₃ 3 ! + I₄ 4 ! - I ₀ =- 1 e 1+1+ 1 2 ! + 1 3 ! + array Let S = _ n =1 ^ I _ n n ! array l - S - I ₀=- 1 e e S =1- I ₀ ~S =1- ₀¹ e ^ - y dy S =1+ [ e ^ - y ]₀¹ ~S =1+ [ 1 e -1 ] S = 1 e array